This document is published in:
International Journal of Impact Engineering
, 2013, 54, 206
216. Doi:
http://dx.doi.org/10.1016/j.ijimpeng.2012.11.003
.
© Elsevier
Finite element analysis of AISI 304 steel sheets subjected to dynamic tension:The
effects of martensitic transformation and plastic strain development on ﬂow
localization
J.A.RodríguezMartínez
a
,
*
,D.Rittel
a
,
b
,R.Zaera
a
,S.Osovski
b
a
Department of Continuum Mechanics and Structural Analysis,University Carlos III of Madrid,Avda.de la Universidad,30,28911 Leganés,Madrid,Spain
b
Faculty of Mechanical Engineering,Technion,32000 Haifa,Israel
o
Keywords:
Dynamic tension
Nec
king
Strain induced martensitic transformation
Critical impact velocity
a b s t r a c t
The paper presents a ﬁnite element study of the dynamic necking formation and energy absorption in
AISI 304 steel sheets.The analysis emphasizes the effects of strain induced martensitic transformation
(SIMT) and plastic strain development on ﬂowlocalization and sample ductility.The material behavior is
described by a constitutive model proposed by the authors which includes the SIMT at high strain rates.
The process of martensitic transformation is alternatively switched on and off in the simulations in order
to highlight its effect on the necking inception.Two different initial conditions have been applied:
specimen at rest which is representative of a regular dynamic tensile test,and specimen with
a prescribed initial velocity ﬁeld in the gauge which minimizes longitudinal plastic wave propagation in
the tensile specimen.Plastic waves are found to be responsible for a shift in the neck location,may also
mask the actual constitutive performance of the material,hiding the expected increase in ductility and
energy absorption linked to the improved strain hardening effect of martensitic transformation.On the
contrary,initializing the velocity ﬁeld leads to a symmetric necking pattern of the kind described in
theoretical works,which reveals the actual material behavior.Finally the analysis shows that in absence
of plastic waves,and under high loading rates,the SIMT may not further increase the material ductility.
2012 Elsevier Ltd.All rights reserved.
1.Introduction
The characterization of the mechanical properties of materials
at high strain rates has become increasingly relevant for the
industry.Accurate knowledge of those properties is usually
required in different engineering applications such as aeronautical
[1,2],automotive [3,4],naval [5,6] and manufacturing [7,8],where
service conditions involve large strains at high strain rates.
Among the experimental characterization tests,the uniaxial
tensile arrangement is deﬁnitely the most commonly used due to
its simplicity.The uniformstate of stress and strain along the gauge
for this test greatly simpliﬁes the interpretation of results.However
the homogeneous deformation eventually ceases due to the onset
of necking in the specimen.Necking in uniaxial stress conditions
has been largely studied since it indicates the onset of a process
which leads to material failure,and therefore determines the
suitability of materials to absorb energy.For quasistatic loading,
Considère [9] showed that in a long and thin bar,the neck develops
at maximumload.Within the framework developed by Hill [10] on
the theory of bifurcation in elasticeplastic solids,different authors
[11,12] demonstrated that the strain at maximum load provides
a lower bound to the strain at which bifurcation occurs;in speci
mens showing large length/width ratio localization starts slightly
after the maximum load whereas in specimens showing small
length/width ratio localization is further delayed.
Under dynamic conditions,the onset of necking is additionally
inﬂuenced by other factors.Different authors concluded that the
condition for instability in tensile tests also depends on the strain
rate sensitivity of the material;Woodford [13] and Ghosh [14]
collected data from tensile tests of a number of metals,showing
a strong delay in necking with increasing logarithmic strain rate
sensitivity.In the meanwhile,a number of theoretical works were
developed to explain this inﬂuence.Hart [15] formulated a stability
criterion for materials exhibiting strain rate sensitivity that was
later reexamined by Ghosh [16].Klepaczko [17] developed
a theoretical framework to include the effect of temperature in the
analysis of instabilities in ratedependent materials.Hutchinson
and Neale [18] concluded that strain rate sensitivity has a strong
* Corresponding author.Tel.:þ34 916248460;fax:þ34 916249430.
Email address:jarmarti@ing.uc3m.es (J.A.RodríguezMartínez).
inﬂuence on the postuniform elongation,retarding necking
localization.In the recent years,a number of papers has been
published which provided further understanding of the interplay
between strain rate sensitivity and necking formation [19e23].
Inertia was the following effect considered as inﬂuential in the
development of dynamic instabilities.Fressengeas and Molinari
[24] used a perturbation analysis to discuss dynamic effects on
ductility,showing that geometrical perturbations are stabilized by
inertia effects.Using a bifurcation analysis,in this case under plain
strain conditions,Shenoy and Freund [25] demonstrated also the
stabilizing effect of inertia.More recently,one should mention
a number of theoretical and numerical works that have provided
additional veriﬁcation of the beneﬁts provided by material inertia
to delay necking formation [19,20,26e29].Material strain hard
ening is certainly an additional factor which enhances ductility.The
Considère condition [9] clearly outlines the favorable effect of
a high strainhardening coefﬁcient in quasistatic conditions,but
this effect has also been addressed by different authors under high
rate loading conditions [16,17,21,29,30].These studies considered
the inﬂuence of parameters,such as material inertia,strain hard
ening and strain rate sensitivity in absence of wave propagation
phenomena.
Among the previous effects modifying the onset of dynamic
necking,strain hardening is known to be inﬂuenced by micro
structural effects such as dynamic phase transformations.Speciﬁ
cally,Strain Induced Martensitic Transformation (SIMT) acts as
a highly potent mechanism of martensite germination associated
with plastic deformation in the austenitic phase.The trans
formation of austenite into martensite is comparable to a dynamic
composite effect due to the progressive appearance of the
martensite during straining,enhancing strain hardening of the
steel.Different authors have recently addressed the role played by
the SIMT in the dynamic behavior of different steel grades [31e34].
It was observed that this phase transformation mechanism can
reasonably be expected to affect the propensity of a material for
dynamic necking,a point that deserves further investigation.
Highspeed hydraulic machines and Split Hopkinson (Kolsky)
Bar devices have been successfully applied to investigate the
mechanical behavior of materials at intermediate to high strain
rates [35,36],the goal being the determination of the intrinsic
properties of the tested material under uniform uniaxial stress
conditions.Therefore different authors introduced original tensile
specimens to optimize the geometry,favor a homogeneous strain
ﬁeld and delay instabilities [37e40].However,imposing a velocity
boundary condition on one side of a solid at rest ewhich is needed
for dynamic testing e necessarily produces a plastic wave front.In
such a case the strain gradients due to the effect of plastic wave
propagation in the specimen may hinder the actual constitutive
behavior of the material.Under large impact velocities,the strain
ﬁeld becomes rapidly nonuniform,affecting the ductility of the
specimen and shifting the position of the neck in the sample.
Hopkinson [41,42] and Hopkinson [43] analyzed the dynamic
loading
of steel
wires and observed that they broke at different
points along their length depending on the loading velocity;the
elastic wave propagation phenomena were considered as respon
sible for this behavior.Furthermore,Von Kàrman and Duwez [44]
and Clark and Wood [45] reported a Critical Impact Velocity (CIV)
such that,when exceeded,the force equilibriumis not fulﬁlled and
necking occurs close to the impacted end with negligible subse
quent plastic strain in the rest of the specimen.The ﬁrst theoretical
work on CIV was proposed by Von Kàrman and Duwez [44],while
Klepaczko [17,46] extended this theory to consider strain rate and
temperature effects.
In order to avoid the drawback of the wave disturbances on
determining material ductility at high strain rates,the ring
expansion test was developed [47] and investigated by different
authors [19,30,48,49].Here,complications resulting from wave
propagation are eliminated until the onset of necking due to the
symmetry of the problem,which implies that the inﬂuence of
loading velocity on material ductility can be studied,virtually,
without limits on the applied velocity.However,the complicated
experimental arrangement required for the ring expansion test
impedes the determination of the material stressstrain charac
teristics.Thus,the uniaxial tensile test,in its dynamic version,
cannot simply be ruled out because of suspected waverelated
effects,at the beneﬁt of the ring expansion test.This remark calls
for an indepth additional evaluation of the dynamic tensile test,in
order to better assess its beneﬁts and also its limitations,while
taking into account the occurrence of dynamic phase trans
formation of the abovementioned kind.
This paper investigates necking formation in AISI 304 steel
sheets subjected to dynamic tension.This steel grade is considered
a reference metastable austenitic stainless steel for studying the
SIMT process at high strain rates since it shows a large amount of
transformed martensite even under adiabatic conditions [50].The
analysis emphasizes the effects of martensitic transformation and
plastic wave propagation on ﬂowlocalization and sample ductility.
For that purpose,ﬁnite element simulations of a dynamic tensile
test have been performed,in which the SIMT has been switched on
and off to disclose the effect of the enhanced strain hardening
produced by the SIMT.In addition,two different initial conditions
have been considered:specimen at rest and specimen with an
initial velocity ﬁeld in the gauge.The analysis shows that inabsence
of an initial velocity ﬁeld and within certain ranges of impact
velocity the neck may be shifted fromthe middle toward the ends
of the specimen.The shifting of the neck is found to be a limiting
factor for the sample ductility.On the contrary,the initial velocity
ﬁeld minimizes the propagation of plastic waves along the longi
tudinal direction of the sample,thus leading to a rather symmetric
necking pattern of the kind described in aforementioned theoret
ical works [19,27].Plastic wave propagation phenomena not only
affects the ductility of the specimen but may also invert the ex
pected increase in energy absorption due to the SIMT.Furthermore,
the role played by the plastic wave propagation hiding the actual
material behavior has been highlighted;and a range of strain rates
for which the measured dynamic tensile characteristics of a mate
rial can be considered as actual material properties has been
determined.Finally the analysis shows that in absence of plastic
waves,and under high loading rates,the SIMT may not provide
further beneﬁts to the material ductility.
The paper is organized as follows.Section 2 provides a brief
summary of the thermoviscoplastic constitutive equations used to
model the mechanical behavior of the AISI 304 steel and empha
sizes the enhanced strain hardening of the material due to
martensitic transformation.Section 3 describes the different ﬁnite
element models developed to perform the study.In Section 4 the
results of the numerical simulations are shown and discussed,
focusing the attention on the effects of martensitic transformation
and plastic wave propagation on ﬂow localization and sample
ductility.The concluding section outlines the main outcomes of this
study.
2.Constitutive modeling of strain induced martensitic
transformation and calculation of effective properties
A complete description of the model,the values of the param
eters identiﬁed for AISI 304 stainless steel and its validation with
dynamic tensile tests results can be found in Zaera et al.[34,51],but
here the key points of the constitutive formulation are further
discussed for completeness.The constitutive description is based
J.A.RodríguezMartínez et al./International Journal of Impact Engineering 54 (2013) 206e216 207
on the previous works of Olson and Cohen [52],Stringfellow et al.
[53] and Papatriantaﬁllou et al.[54],all developed to account for
SIMT in steels containing metastable austenite,and includes
modiﬁcations in the following items:
e According to Olson and Cohen [52] the intersection of shear
bands in austenite is considered as the dominant mechanismof
SIMT.The kinetics of the transformation is described by an
exponential expression in which the plastic deformation in
austenite is multiplied by a coefﬁcient
a
favoring the shear
band deformation mode.The parameter
a
decreases with
temperature,as observed by Olson and Cohen [52].The
following phenomenological law is proposed to ﬁt this
dependence
a
ð
Q
Þ ¼
a
0
h
1
Q
exp
a
1
1
Q
1
i
(1)
where
a
0
and
a
1
are constants and
Q
is a normalized temper
ature.In comparison with the polynomial expressions applied
by other authors to describe the change of
a
with temperature
[52,55,56],the one proposed in this work facilitates to capture
the decrease of the transformation rate with increasing
temperature within the temperature range in which the SIMT
occurs.
e The linearly temperature dependent normalized thermody
namic driving force for the martensitic transformation,
proposed by Stringfellow et al.[53],is replaced here by an
exponential equation.This new law provides a greater
temperature sensitivity of the rate of martensitic trans
formation.This becomes relevant at high strain rates where the
temperature rise of the material results from the adiabatic
character of the plastic deformation.According to the experi
mental results reported by RodríguezMartínez et al.[50] at
high deformation rates,and therefore signiﬁcant temperature
increase,small temperature variations lead to signiﬁcant
differences on the volume fraction of martensite formed.
e The thermal deformation tensor rate is included in the gener
alized Hooke’s lawfor hypoelasticplastic materials,taking into
account that the adiabatic increase of temperature may lead to
nonnegligible thermal strains.The rate of thermal deforma
tion tensor is assumed to be isotropic
d
q
¼
a
q
_
q
1 (2)
a
q
being the coefﬁcient of thermal expansion,
q
the absolute
temperature and 1 the second order unit tensor.Mahnken et al.
[57] used a rule of mixtures to calculate
a
q
taking into account
the different coefﬁcients for thermal expansion of martensite
and austenite.However,different authors [58] shown that the
thermal expansioncoefﬁcient does not obeythelawof mixtures
because a state of microstresses appears between the phases
when submitted to a temperature increase.Therefore the
authors have chosen a constant and characteristic value of
a
q
.
e The thermal softening of each solid phase is considered in the
model through a thermoviscoplastic potential,which has been
deﬁned in such a way that the yield stress follows a power
thermoviscoplastic law.These laws are commonly accepted to
model strain/strain rate hardening and temperature softening
[19,59e61].The effective properties of the thermoviscoplastic
heterogeneous material are calculated using a plastic potential
including viscous and thermal effects for both austenite and
martensite,using the modiﬁed secant method proposed by
Suquet [62e64] and the solution algorithm developed by
Papatriantaﬁllou et al.[54].
e Anoriginal thermodynamic scheme to capture the variability of
the TayloreQuinney coefﬁcient in austenitic steels showing
strain induced martensitic transformation at high strain rates is
included.Different heat sources involved in the temperature
increase of the material are taken into account.These are the
latent heat released due to the exothermic character of the
transformation and the heat released due to austenite and
martensite straining.The intrinsic dissipation in each of
the constitutive phases is computed as a constant fraction of
the corresponding plastic work power.The variability of the
TayloreQuinney coefﬁcient stems from the evolution of the
phase fractions and from the latent heat released upon phase
transformation.Through a differential treatment of these
dissipative terms,the TayloreQuinney coefﬁcient develops
a direct connection with the martensitic transformation,and
the proposed model strives to provide complementary insight
regarding heat generation in dynamically phase transforming
alloys,in which the heat power might be greater than the
intrinsic dissipation power.
The socalled dynamic composite effect due to the progressive
appearance of martensite is clearly shown in Fig.1.The yield
strength of martensite is higher than that of austenite,increasing
the ﬂow stress and strain hardening in the steel grade.
3.Finite element modeling of dynamic tension
Dynamic tension FE simulations are conducted in order to
evaluate the effects of SIMT and plastic strain development on the
energy absorption capabilities of the material.The simulations are
run for strain rates ranging between 500 s
1
_
3
20;000 s
1
.The
authors are aware that this range of loading rates exceeds the
regular experimental capabilities,however this way of proceeding
is efﬁcient to analyze the effect of the aforementioned mechanical
factors on the strain localization process.Two different numerical
conﬁgurations are addressed:full sample description and single
element simulations.
3.1.Full sample description
The geometry of the tensile specimen is taken fromRodríguez
Martínez et al.[50],Fig.2.The mesh consists of 10,941 eightnode
σ
ε
Fig.1.Comparison between numerical predictions of the stressestrain behavior for
AISI 304 grade,in which the martensitic transformation has been switched on and off
_
3
¼ 500 s
1
.
J.A.RodríguezMartínez et al./International Journal of Impact Engineering 54 (2013) 206e216208
trilinear brick elements with reduced integration.The integral
viscoelastic approach available in ABAQUS/Explicit [65] has been
used to prevent hourglass deformation modes,with the scale factor
used for all hourglass stiffnesses being equal to one.According to
the considerations reported by Zukas and Scheffer [66],the gauge
of the specimen has been meshed using elements whose aspect
ratio was close to 1:1:1 (z1/3 1/3 1/3 mm
3
).Loading in the
specimen has been introduced through the following boundary
condition:an axial velocity V instantaneously applied on side A e
see Fig.2 e which remains constant throughout the entire process
[40,67].
Two different initial conditions have been applied:
e Noﬁeld:The initial condition is V ¼ 0.The application of this
initial condition to a specimen initially at rest leads to the
propagation of a plastic wave front along the longitudinal axis
of the sample [21,68].This conﬁguration is representative of
a typical experimental setup.
e Field:The axial velocity along the length of the sample V
1
is
initialized as schematically described in Fig.3.
This initial condition avoids the abrupt motion of the sample
along the axial direction at t ¼ 0.Here should be noted that the
initial velocity along directions X
2
and X
3
is set to 0.The authors are
aware that this leads to acceleration of the sample along X
2
and X
3
directions at the beginning of loading.Moreover,let us note that
since the reference conﬁguration is stressfree at the onset of
loading the sample is subjected to an initial stress increase as
a consequence of the load application.However,it is reasonable to
assume that the distortion of the strain ﬁeld along the longitudinal
axis of the specimen is the dominant mechanism governing the
process of necking formation.The potential strain disturbances
along X
2
and X
3
directions and the stress increase at the beginning
Fig.2.Schematic representation of the 3D model.Mesh,dimensions (mm),boundary conditions and loading condition.
Fig.3.Schematic representation of the initial velocity ﬁeld applied to the sample in
the ﬁeld simulations.
J.A.RodríguezMartínez et al./International Journal of Impact Engineering 54 (2013) 206e216 209
of loading may play a secondary role in the neck inception.In
forthcoming sections of this paper the effectiveness of this initial
condition in minimizing the propagation of the plastic wave front is
strengthened.
It is worth noting that in order to mimic most of the existing
experimental tests,no initial numerical or geometrical imperfec
tion has been assumed.A regular mesh was used to minimize
meshrelated imperfections,so that the necking instability would
develop froman initially smooth surface.This procedure has been
shown effective in reproducing the localization mechanics of the
dynamic tension problemas demonstrated by Rusinek et al.[40].
3.2.Single element conﬁguration
FE models with a single element having one quadrature point
allowefﬁcient integration of the constitutive models under partic
ular loading conditions.The effect of a given specimen geometry is
discarded and the results yield the speciﬁc response of the consti
tutive models.Stressstrain characteristics obtained from this
conﬁguration are used as a reference for evaluation of the results
obtained from the full sample simulations.One 1 1 1 mm
3
8node trilinear brick element with reduced integration was used
inthesimulations.Fig.4shows theloadingandboundaryconditions
imposedtothe element todeﬁne the tensile stress state.It shouldbe
noted that for a single element having one integration point the
generation of a plastic wave front is precluded.There is no upper
limit in velocity which impedes determination of the stressstrain
material characteristics,i.e.the critical impact velocity is avoided
in this approach.
4.Results and analysis
The reported stressstrain curves are obtained from the full
sample numerical simulations following the regular procedure
used in experimentation.The sample displacement (used to
determine the strain in the sample gauge during the simulations) is
recorded at the impacted side of the specimen and the force (used
to determine the stress) at the clamped end e see Fig.2.Moreover,
similarly to Xue et al.[21] and RodríguezMartínez et al.[69],the
localized necking strain
3
neck
e fromthis point on denoted simply
by necking strain e is determined by the condition du
1
/dt ¼ 0,
where u
1
is the longitudinal displacement in the X
1
direction
measured at a node beside the necked zone,in the direction toward
the clamped side,and t refers to time,Fig.5.Once the necking strain
is known,it can be used as the upper limit of integration of the
stressestrain curves for the determination of the energy absorbed
by the sample until strain localization occurs,E
s
.
4.1.The role of SIMT in presence of plastic waves
The role of SIMT on the energy absorption capability of the
material in presence of plastic waves is evaluated using noﬁeld
initial conditions.For that task,the transformation has been
systematically switched on and off.Fig.6 shows the necking strain
3
neck
and the energy absorbed by the sample E
s
as a function of the
loading rate _
3
for simulations in which the SIMT has been activated
and deactivated.Note that 10% is the percentagevalue of the data
considered,which is represented by the error bars.
Consider ﬁrst the general behavior of
3
neck
and E
s
as a function of
the strain rate _
3
.For this,we will just refer to simulations in which
the SIMT has been switched on e solid symbols in Fig.6.It can be
observed that for strain rates up to w3000 s
1
the necking strain
and the energy absorbed by the sample slightly increase with
impact velocity.This comprises the range of loading rates for which
necking takes place in the middle of the sample,Fig.6.Then,within
Fig.4.Schematic representation of the single element model.Boundary and loading
conditions.
Fig.5.Schematic representation of necked zone showing the measurement point used
for determination of the localized necking condition.
ε
ε
ε
ε
Fig.6.Necking strain
3
neck
and energy absorbed by the sample E
s
as a function of the
loading rate _
3
for noﬁeld simulations in which the martensitic transformation has
been switched on and off.
J.A.RodríguezMartínez et al./International Journal of Impact Engineering 54 (2013) 206e216210
certain range of strainrates,upto 4500 s
1
approximately,
3
neck
and
E
s
slightly decrease with impact velocity.This behavior corresponds
to the loading rates for which the neck takes place closer to the
clamped side,Fig.7.Whenthe neck is inceptedclose tothe clamped
side,the shoulder of the specimen limits the development of plastic
deformation inducing large strain gradients responsible for
decreasing
3
neck
.At higher strain rates,the necking location starts
moving from the clamped side to the impacted end leading to
a continuous rise in
3
neck
and E
s
with impact velocity which extends
until attainment of the CIV at w150 m/s (which corresponds to
7500 s
1
),Fig.6.Then,the sample ductility continuously decreases
with loading rate.Here it should be noted that in order to observe
a drastic reduction in ductility beyond the CIV,it is necessary to
analyze specimens showing larger length/width and length/thick
ness ratios,as pointed out by Von Kàrman and Duwez [44].It is
worth noting that identical relationships between
3
neck
,E
s
and _
3
are
observed in the case of deactivated SIMT;the only difference lies in
the loading velocity values for which the neck takes place in the
middle of the gauge,in the clamped side or in the impacted side,
Fig.7.In the case of noSIMT,the shift of the neck (both variants,i.e.
necking closer to the clamped side,necking closer to the impacted
side) occurs at lower impact velocities due to lower material ﬂow
stress and strain hardening.The calculated results concerning the
general dependence of necking strain and energy absorbed by the
sample on impact velocity and necking location are all well docu
mented in the literature [40,45,67].This clearly supports the idea
that necking location and necking strain are direct consequence of
test loading rate,material constitutive behavior and plastic wave
propagation (in addition to other factors related to the geometry
and dimensions of the specimen,material density) and,especially
at high strain rates,cannot be considered the result of a random
process governed by material or geometrical defects.
Next,let us identify the contribution of the SIMT to the calculated
values of
3
neck
andE
s
.At lowstrainrates,uptow4000s
1
,thenecking
strain and the energy absorbed by the sample until necking are
slightly larger if the SIMT is active.This behavior could be expected
since the martensitic transformation increases the material strain
hardening,the latter being a variable involved in necking retardation
[21].It is worth noting that the relationshipbetweenimproved strain
hardening and necking retardation has been frequently presented as
universal in the literature.However this is not the case for boundary
value problems involving plastic wave propagation.This is illustrated
inFig.6 for the range of strain rates 4000 s
1
< _
3
< 6000 s
1
.Within
these loading rates both necking strain and energy absorbed by the
sample are larger when the martensitic transformation is switched
off.This unexpected behavior,barely reported in the literature to the
authors knowledge,is caused by the plastic wave front triggered by
the abrupt motion of the impacted side at t ¼ 0.In other words,the
plastic wave propagation at the onset of loading inﬂuences the
necking position,and thus the ductility of the sample as discussed in
theprevious paragraph.Aclear illustrationof theinﬂuenceof necking
position on the sample ductility is reported in Fig.8,where the
transverse displacement of the specimen in the X
2
direction is
depicted as a function of the normalized gauge length.For
_
3
¼ 4000 s
1
and _
3
¼ 5000 s
1
the neck takes place closer to the
clamped end when the SIMT is switched on.Furthermore,for these
loadingrates,theneckismoredevelopedinthecaseof activatedSIMT.
In other words,the maximumvertical displacement is larger in the
case of activated SIMT.For 6000 s
1
< _
3
< 10;000 s
1
the neck takes
place close to the impacted end independently on the martensitic
transformation,leading to larger values of
3
neck
and E
s
if the SIMT is
active.Finallyit is worthnotingthat for strainrates
_
3
a10;000 s
1
the
CIV has been largely exceeded for both,SIMT switched on and SIMT
Fig.7.Illustration of the interplay between initial strain rate _
3
and necking location.
μ
μ
(a)
(b)
Fig.8.Vertical displacement of the gauge length as a function of the normalized gauge
length for simulations in which the SIMT has been alternatively switched on and off.
(a) Strain rate _
3
¼ 4000 s
1
,loading time t ¼ 175.2
m
s.(b) Strain rate _
3
¼ 5000 s
1
,
loading time t ¼ 150.0
m
s.
J.A.RodríguezMartínez et al./International Journal of Impact Engineering 54 (2013) 206e216 211
switched off simulations and the energy absorbed by the sample
becomes independent of the material description.
4.2.The role of plastic waves
The role of plastic wave propagation on the onset of necking is
evaluated through the comparison of ﬁeld and noﬁeld simulations
in which the SIMT is active.This analysis is split into two parts.
Firstly,we analyze the different results obtained fromthe ﬁeld and
noﬁeld simulations in full sample description simulations and
their comparison with the single element computations.The
second part addresses speciﬁcally the material response in the ﬁeld
simulations.
e Fig.9 shows the necking strain
3
neck
and the energy absorbed
by the sample E
s
as a function of the loading rate _
3
for ﬁeld and
noﬁeld simulations.Within the range of strain rates
_
3
< 3000 s
1
the results obtained using both initial velocity
conditions are practically identical.In both cases the neck is
incepted in the middle of the sample.Within the range
3000 s
1
<
_
3
< 7500 s
1
the difference arises and both
3
neck
and E
s
are signiﬁcantly larger in the case of the ﬁeld simula
tions.Then,in the case of noﬁeld simulations the neck occurs
in the clamped end and in the case of ﬁeld computations the
neck still occurs in the middle of the sample.Next,within the
range 7500 s
1
< _
3
< 10;000 s
1
the trend is inverted and
3
neck
and E
s
are larger in the case of noﬁeld calculations.For ﬁeld
simulations,a transition to two evenly spaced necks appears as
strain rate increases,leading to a transient decrease in the
energy required for necking formation,as will be explained
later.In the case of noﬁeld simulations,the neck is nucleated
in the impacted end leading to the CIV.It can be stated that as
soon the neck leaves the middle of the sample in the noﬁeld
computations,the difference between ﬁeld and noﬁeld
simulations sets in.As mentioned earlier,the plastic wave
front generated by the impact in the case of the noﬁeld
simulations is responsible for such behavior.Consequently,at
strain rates above 9000 s
1
,the necking strain for the noﬁeld
ε
ε
ε
ε
Fig.9.Necking strain
3
neck
and energy absorbed by the sample E
s
as a function of the
loading rate _
3
for SIMT simulations in which the initial velocity ﬁeld has been switched
on and off.
Fig.10.(a) Plastic strain upon the normalized gauge length at 10,000 s
1
for noﬁeld and ﬁeld simulations in which the SIMT has been switched on.(b) Contours of plastic strain at
10,000 s
1
for noﬁeld and ﬁeld simulations in which the SIMT has been switched on.
J.A.RodríguezMartínez et al./International Journal of Impact Engineering 54 (2013) 206e216212
calculation keeps lower than that corresponding to the ﬁeld
case.Fig.10a shows the plastic strain along the normalized
gauge length in the case of _
3
¼ 10;000 s
1
for both initial
velocity conditions and different loading times.It can be
observed that if the velocity ﬁeld is not initialized,the plastic
strain concentrates close to the impacted side from the very
beginning of loading similarly to the solution proposed by Von
Kàrman and Duwez [44].On contrary,if the velocity ﬁeld is
initialized,the plastic strain in the sample at the beginning of
loading is largely constant.This conﬁrms the role of the initial
velocity ﬁeld minimizing the plastic wave propagation.In
addition,the results clearly indicate that the plastic waves,and
therefore the necking location,distort the necking strain of
the material.Figs.11 and 12 show the comparison between
stressestrain curves obtained from the full sample computa
tions and the single element simulations for ﬁeld and noﬁeld
conditions at two different initial strain rates,
_
3
¼ 4000 s
1
and
_
3
¼ 7500 s
1
.It can be observed that the stressestrain
characteristics obtained fromthe noﬁeld simulations and the
single element results do not match,especially at low strain
levels.This difference increases with the impact velocity.In
other words,as soon as the neck does not take place in the
middle of the gauge the measured material characteristic in the
noﬁeld simulations is not representative of the real material
behavior.This ﬁnding demonstrates that the velocity which
imposes an upper limit to the dynamic tension test for deter
mining material the actual stressestrain curve of the material is
not the CIV but the velocity which causes the necking to move from
the center of the sample to the clamped end.Beyond this limit,
the displacement recorded at the impacted side and the force
recorded at the clamped side do not correspond to a uniform
state of stress and strain along the specimen,as needed for
a proper interpretation of the experimental measurements.
This observation is relevant for experimental work,and the
results shown here were not previously reported to the best of
the authors knowledge.
e Consider now the specimen response in the ﬁeld simulations.
Froma general point of view,the results shown in Fig.9 reveal
that in the absence of plastic wave propagation,both
3
neck
and
E
s
increase with loading rate e this is not true within
4500 s
1
< _
3
< 7500 s
1
as discussed later.This behavior
seems to be explained by the role played by strain rate _
3
on
necking retardation [19,23,29].As expected,the CIV is avoided
in absence of plastic waves.Moreover,it is worth nothing that
for the lower strain rates considered,up to w7500 s
1
,a single
neck located in the middle of the gauge is formed.Then,this
single neck suddenly gives way to the nucleation of two
(rather) evenly spaced necks along the gauge of the sample,
Fig.10,that are also found at higher strain rates.Following
RodríguezMartínez et al.[69] this transition may be related to
material inertia aspects;taking into account that the term
inertia not only covers material density but it also accounts for
the intrinsic effects that sample dimensions,ﬂow stress level
and loading rate all have on necking inception as described
elsewhere [19,23,29].This may explain the nucleation of two
regularly spaced necks assuming that at w7500 s
1
,the elon
gation of the sample before necking is such that it permits the
σ
ε
σ
ε
(a)
(b)
Fig.11.Comparison between stressestrain curves for simulations using the full sample
description and the single element.SIMT is switched on.Loading rate 4000 s
1
.(a) No
ﬁeld simulation,(b) ﬁeld simulation.
σ
ε
σ
ε
(a)
(b)
Fig.12.Comparison between stressestrain curves for simulations using the full
sample description and the single element.SIMT is switched on.Loading rate 7500 s
1
.
(a) Noﬁeld simulation,(b) ﬁeld simulation.
J.A.RodríguezMartínez et al./International Journal of Impact Engineering 54 (2013) 206e216 213
promotion of a more favorable wavelength mode for the
necking formation.This more favorable wavelength should to
be at least twice shorter than the stretched gauge allowing the
formation of two necks.The transition to a more favorable
wavelength is accompanied by a decrease in the energy
required for necking formation which may explain the tran
sient decrease inthe necking strainreported in Fig.9 withinthe
range 4500 s
1
<
_
3
< 7500 s
1
.Moreover,it was previously
noted that when the necks are incepted close to the shoulders
of the sample,the development of the plastic deformation gets
limited,which can act as an additional factor that decreases the
necking strain within this range of loading rates.It should be
highlighted that while the transition from the formation of
a single neck to the formation of two necks in the ﬁeld simu
lations is dictated by material inertia aspects,the necking
location in the noﬁeld simulations was dictated by the plastic
wave propagation.Fig.13 illustrates the vertical displacement
of the gauge length as a function of the normalized gauge
length for different ﬁeld simulations.Three selectedstrainrates
are 7500 s
1
,12,500 s
1
and 15,000 s
1
,for which two necks
were nucleated.The loading time has been set in order to
obtain the same maximumvertical displacement for the three
cases considered as indicated in Fig.13.For the lower loading
rate considered,the incepted necks develop in a rather
different manner,Fig.13.It has been detected that one of them
e the one closer to the clamped side e develops ﬁrst in time
may be because of the disturbances resulting fromthe sudden
rise in stress experienced by the sample at the onset of loading
e see Section 4.a.Once the ﬁrst neck is formed an unloading
wave emanates from the necked region slowing down the
growth rate of the secondary neck [70].This explanation ﬁnds
agreement with the original Mott’s theory [71] and the
subsequent developments by Grady and coworkers [72e74].
Moreover it can be observed that the two necks thus formed,
become increasingly symmetrical as the loading rate increases,
i e.as the role played by inertia increases.This behavior agrees
with theoretical and numerical observations reported else
where for other multiple necking problems [22,27,69].
4.3.The role of SIMT in absence of plastic waves
The role of SIMT in absence of plastic waves on the energy
absorption capability of the material is evaluated using ﬁeld initial
conditions.Fig.14 shows the necking strain
3
neck
and the energy
absorbed by the sample E
s
as a function of the loading rate
_
3
for
simulations in which the SIMT has been activated and deactivated.
It can be observed that now the picture is completely different to
that reported in Fig.6;the relative inﬂuence of the SIMT on the
material response is found to be highly dependent on the presence
or absence of the plastic waves.
It is worth noting that the recorded values of
3
neck
seem to be
rather independent of the martensitic transformation,in fact in
some cases
3
neck
is larger in the case of deactivating the SIMT.The
explanation may be related to the emerging role played by inertia e
the concept of inertia was mentioned in Section 4.a eat high strain
rates [19,29,69].In other words,the stabilizing effect of the
enhanced strain hardening provided by the SIMT seems to be
balanced by the destabilizing effect of its associated increase in ﬂow
stress [23].However,although the necking strain values are quite
similar for both conditions,the computations in which the SIMT is
switched on show larger values of E
s
;it has to be noted that such
difference tends to decrease with the increasing loading rate.In
other words,the SIMT does not provide enhanced ductility to the
material but it is still beneﬁcial in terms of energy absorption due to
the larger material ﬂowstress.This calls for a reassessment of the
usefulness of the SIMT for enhancing material ductility in high
strain rate applications,for which the material response may be
governed by inertia effects.
5.Conclusions
In this paper the processes of strain localization and necking
formation in AISI 304 steel sheets subjected to dynamic tension
have been investigated using ﬁnite element simulations.Two
different numerical conﬁgurations are addressed:single element
simulations and full specimen geometry simulations.In the full
specimen geometry calculations two different initial conditions
have been applied;noﬁeld which is representative of a regular
experimental arrangement and ﬁeld which allows minimizing the
propagation of plastic waves along the longitudinal direction of the
sample.The material behavior is described by a constitutive model
proposed by the authors which includes explicitly the SIMT,thus
characterizing the response of the AISI 304 at high loading rates.
The analysis is focused on the effects of SIMT and plastic wave
propagation on the process of ﬂow localization,which in turn
ε
ε
ε
ε
Fig.14.Necking strain
3
neck
and energy absorbed by the sample E
s
as a function of the
loading rate
_
3
for ﬁeld simulations in which the martensitic transformation has been
switched on and off alternatively.
μ
μ
μ
Fig.13.Vertical displacement of the gauge length as a function of the normalized
gauge length for ﬁeld simulations in which the SIMT has been switched on.Three
different loading rates are illustrated:7500 s
1
,12,500 s
1
and 15,000 s
1
.The loading
time has been set in order to obtain the same maximumvertical displacement for the
three cases considered.
J.A.RodríguezMartínez et al./International Journal of Impact Engineering 54 (2013) 206e216214
determines the energy absorption capacity of the material at high
loading rates.
The main conclusions that emerge fromthis work are as follows:
e In presence of plastic waves,and under certain strain rate
conditions,the increased strain hardening provided by the
SIMT process may not delay the onset of necking,thus limiting
the energyabsorption capacity of the material.This unexpected
behavior is barely reported in the literature,and it is related to
the motion of the necking location along the gauge length.
e Once a certain impact velocity is exceeded,the presence of
plastic waves in the dynamic tensile test hinders the actual
material behavior.In other words,as soon as the neck does not
take place in the middle of the gauge,the stressestrain curve
obtained fromdynamic tensile tests is not representative of the
actual material behavior.This ﬁnding demonstrates that the
velocity which imposes an upper limit to the dynamic tension
test for determining material properties is not the CIV,but the
velocity which causes the necking to move fromthe center of
the sample.
e In the absence of plastic waves,and under certain strain rate
conditions,the SIMT may not provide the anticipated enhanced
material ductility.The stabilizing effect of the enhanced strain
hardening provided by the SIMT is balanced by the destabi
lizing effect of its associated increase in ﬂow stress.This
behavior seems to be a limiting factor to the homogeneous
deformation behavior of the material at high strain rates.This
observation calls for a reassessment of the beneﬁcial effects of
SIMT at high loading rates,for which the material response
may be governed by inertial effects.
Acknowledgments
J.A.RodríguezMartínez and R.Zaera express sincere gratitude
to Dr.Guadalupe Vadillo,Professor José FernándezSáez and
Professor Alain Molinari for helpful discussions on the role played
by material aspects on the formation of plastic instabilities in
ductile materials subjected to high strain rates.
D.Rittel acknowledges the support of Carlos III University with
a Cátedra de Excelencia funded by Banco Santander during academic
year 2011e2012.
The researchers of the University Carlos III of Madrid are
indebted to the Comunidad Autónoma de Madrid (Project CCG10
UC3M/DPI5596) and to the Ministerio de Ciencia e Innovación de
España (Project DPI/200806408) for the ﬁnancial support received
which allowed conducting part of this work.
References
[1] Karagiozova D,Mines R.Impact of aircraft rubber tyre fragments on
aluminiumalloy plates:IInumerical simulation using LSDYNA.International
Journal of Impact Engineering 2007;34:647e67.
[2] Varas D,ZaeraR,LópezPuenteJ.Numerical modellingof thehydrodynamic ram
phenomenon.International Journal of Impact Engineering 2009;36:363e74.
[3] Rusinek A,Zaera R,Forquin P,Klepaczko JR.Effect of plastic deformation and
boundary conditions combined with elastic wave propagation on the collapse
site of a crash box.ThinWalled Structures 2008;46:1143e63.
[4] Kazanci Z,Bathe K.Crushing and crashing of tubes with implicit time inte
gration.International Journal of Impact Engineering 2012;42:80e8.
[5] Wang L,Yang L,Huang D,Zhang Z,Chen G.An impact dynamics analysis on
a newcrashworthy device against shipebridge collision.International Journal
of Impact Engineering 2008;35:895e904.
[6] Ehlers S.The inﬂuence of the material relation on the accuracy of collision
simulations.Marine Structures 2010;23:462e74.
[7] Miguelez MH,Zaera R,Molinari A,Cheriguene R,Rusinek A.Residual stresses
in orthogonal cutting of metals:the effect of thermomechanical coupling
parameters and of friction.Journal of Thermal Stresses 2009;32:269e89.
[8] Verleysen P,Peirs J,Slycken JV,Faes K,Duchene L.Effect of strain rate on the
forming behaviour of sheet metals.Journal of Materials Processing Technology
2011;211:1457e64.
[9] Considère AG.L’emploi du fer de l’acier dans les constructions.Ann Ponts et
Chausses 1885;9:574e5.
[10] Hill R.A general theory of uniqueness and stability in elasticplastic solids.
Journal of the Mechanics and Physics of Solids 1958;6:236e49.
[11] Miles J.Bifurcation in plastic ﬂow under uniaxial tension.Journal of the
Mechanics and Physics of Solids 1971;19:89e102.
[12] Hutchinson J,Miles J.Bifurcation analysis of the onset of necking in an elastic/
plastic cylinder under uniaxial tension.Journal of the Mechanics and Physics
of Solids 1974;22:61e71.
[13] Woodford D.Strain rate sensitivity as a measure of ductility.Transactions of
the American Society for Metals 1969;62:291e3.
[14] Ghosh AK.The inﬂuence of strain hardening and strainrate sensitivity on
sheet metal forming.Transactions ASME Journal of Engineering Materials
Technology 1977;99:264e74.
[15] Hart EW.A theory for ﬂowof polycrystals.Acta Metallurgica 1967;15:1545e9.
[16] Ghosh AK.Tensile instability and necking in materials with strain hardening
and strainrate hardening.Acta Metallurgica 1977;25:1413e24.
[17] Klepaczko JR.Generalized conditions for instability in tension.International
Journal of Mechanical Sciences 1968;10:297e313.
[18] Hutchinson JW,Neale KW.Inﬂuence of strain rate sensitivity on necking
under uniaxial tension.Acta Metallurgica 1977;25:839e46.
[19] Mercier S,Molinari A.Analysis of multiple necking in rings under rapid radial
expansion.International Journal of Impact Engineering 2004;30:403e19.
[20] Zhou F,Molinari JF,Ramesh KT.An elastoviscoplastic analysis of ductile
expanding ring.International Journal of Impact Engineering 2006;33:880e91.
[21] Xue Z,Vaziri A,Hutchinson JW.Material aspects of dynamic neck retardation.
Journal of the Mechanics and Physics of Solids 2008;56:93e113.
[22] Mercier S,
Granier N,Molinari A,Llorca F,Buy F.Multiple necking during the
dynamic expansion of hemispherical metallic shells,from experiments to
modelling.Journal of the Mechanics and Physics of Solids 2010;58:955e82.
[23] Vadillo G,RodríguezMartínez JA,FernándezSáez J.On the interplay between
strain rate and strain rate sensitivity on ﬂow localization in the dynamic
expansion of ductile rings.International Journal of Solids and Structures 2012;
49:481e91.
[24] Fressengeas C,Molinari A.Inertia and thermal effects on the localization of
plastic ﬂow.Acta Metallurgica 1985;33:387e96.
[25] Shenoy VB,Freund LB.Necking bifurcations during high strain rate extension.
Journal of the Mechanics and Physics of Solids 1999;47:2209e33.
[26] Pandolﬁ A,Krysl P,Ortiz M.Finite element simulation of ring expansion and
fragmentation:the capturing of length and time scales through cohesive
models of fracture.International Journal of Fracture 1999;95:297.
[27] Sørensen NJ,Freund LB.Unstable neck formation in a ductile ring subjected to
impulsive radial loading.International Journal of Solids and Structures 2000;
37:2265e83.
[28] Becker R.Ring fragmentation predictions using the Gurson model with
material stability conditions as failure criterion.International Journal of Solids
and Structures 2002;39:3555e80.
[29] Mercier S,Molinari A.Predictions of bifurcations and instabilities during
dynamic extensions.International Journal of Solids and Structures 2003;40:
1995e2016.
[30] Rusinek A,Zaera R.Finite element simulation of steel ring fragmentation
under radial expansion.International Journal of Impact Engineering 2007;34:
799e822.
[31] Rusinek A,Klepaczko JR.Experiments on heat generated during plastic defor
mationandstoredenergy for TRIPsteels.Materials andDesign2009;30:35e48.
[32] RodríguezMartínez JA,Pesci R,Rusinek A,Arias A,Zaera R,Pedroche DA.
Thermomechanical behaviour of TRIP 1000 steel sheets subjected to low
velocity perforation by conical projectiles at different temperatures.Interna
tional Journal of Solids and Structures 2010;47:1268e84.
[33] RodríguezMartínez JA,Rusinek A,Pesci R.Experimental survey on the
behaviour of AISI 304 steel sheets subjected to perforation.ThinWalled
Structures 2010;48:966e78.
[34] Zaera R,RodríguezMartínez JA,Casado A,FernándezSáez J,Rusinek A,
Pesci R.A constitutive model for analyzing martensite formation in austenitic
steels deforming at high strain rates.International Journal of Plasticity 2012;
29:77e101.
[35] Gray III GT.Classic splitHopkinson pressure bar testing.In:Kuhn H,Medlin D,
editors.ASM handbook.Mechanical testing and evaluation,Vol.8.Materials
Park,OH:ASM International;2000.p.462e76.
[36] Field J,Walley S,Proud W,Goldrein H,Siviour C.Review of experimental
techniques for high rate deformation and shock studies.International Journal
of Impact Engineering 2004;30:725e75.
[37] Staab GH,Gilat A.A directtension split Hopkinson bar for high strainrate
testing.Experimental Mechanics 1991;31:232e5.
[38] Tanimura S,Mimura K.Newly developed dynamic testing methods and
dynamic strength of some structural materials.vol.1 of Impact Eng.Appl.
London:Proceeding of the 4th International Symposium on Impact Engi
neering;2001.p.57e64.
[39] Huh H,Kang WJ,Han SS.A tension split Hopkinson bar for investigating the
dynamic behavior of sheet metals.Experimental Mechanics 2002;42:8e17.
[40] Rusinek A,Zaera R,Klepaczko JR,Cheriguene R.Analysis of inertia and scale
effects on dynamic neck formation during tension of sheet steel.Acta Mate
rialia 2005;53:5387e400.
[41] Hopkinson J.On the rupture of iron wire by a blow.Proceedings of the
Manchester Literary and Philosophical Society 1872;11:40e5.
J.A.RodríguezMartínez et al./International Journal of Impact Engineering 54 (2013) 206e216 215
[42] Hopkinson J.Further experiments on the rupture of an iron wire.Proceedings
of the Manchester Literary and Philosophical Society 1872;11:119e21.
[43] Hopkinson B.The effects of momentary stresses in metals.Proceedings of the
Royal Society of London 1905;74:498e506.
[44] Von Kàrman T,Duwez P.The propagation of plastic deformation in solids.
Journal of Applied Physics 1950;21:987e94.
[45] Clark DS,Wood DS.The inﬂuence of specimen dimension and shape on the
results in tension impact testing.The American Society of Mechanical Engi
neers 1950:577e85.
[46] Klepaczko JR.Review on critical impact velocities in tension and shear.
International Journal of Impact Engineering 2005;32:188e209.
[47] Niordson FL.A unit for testing materials at high strain rates.Experimental
Mechanics 1965;5:29e32.
[48] Hu X,Daehn GS.Effect of velocity on ﬂow localization in tension.Acta
Materialia 1996;44:1021e33.
[49] Zhang H,RaviChandar K.On the dynamics of necking and fragmentation e I.
realtime and postmortem observations in Al 6061O.International Journal
of Fracture 2006;142:183e217.
[50] RodríguezMartínez JA,Pesci R,Rusinek A.Experimental study on the
martensitic transformation in AISI 304 steel sheets subjected to tension under
wide ranges of strain rate at room temperature.Materials Science and Engi
neering:A 2011;528:5974e82.
[51] Zaera R,RodríguezMartínez JA,Rittel D.On the TayloreQuinney coefﬁcient in
dynamically phase transforming materials.Application to 304 stainless steel.
International Journal of Plasticity 2013;40:185e201.
[52] Olson GB,Cohen M.Kinetics of straininduced martensitic nucleation.
Metallurgical Transactions A 1975;6A:791e5.
[53] StringfellowRG,Parks DM,Olson GB.A constitutive model for transformation
plasticity accompanying straininduced martensitic transformations in
metastable austenitic steels.Acta Metallurgica and Materialia 1992;40(7):
1703e16.
[54] Papatriantaﬁllou I,Aravas N,Haidemenopoulos G.Finite element modelling of
TRIP steels.Steel Research International 2004;75:732e8.
[55] Tomita Y,Iwamoto T.Constitutive modelling of trip steel and its application to
the improvement of mechanical properties.International Journal of
Mechanical Sciences 1995;37:1295e305.
[56] Iwamoto T,Tsuta T.Computational simulation on deformation behavior of CT
specimens of TRIP steel under mode I loading for evaluation of fracture
toughness.International Journal of Plasticity 2002;11:1583e606.
[57] Mahnken R,Schneidt A,Antretter T.Macro modelling and homogenization for
transformation induced plasticity of a lowalloy steel.International Journal of
Plasticity 2009;25:183e204.
[58] Sideridis E,Kytopoulos VN,Kyriazi E,Bourkas G.Determination of thermal
expansion coefﬁcient of particulate composites by the use of a triphase model.
Composites Science and Technology 2005;65:909e19.
[59] Klepaczko JR.A practical stressstrainstrain ratetemperature constitutive
relation of the power form.Journal of Mechanical Working Technology 1987;
15:143e65.
[60] Molinari A,Musquar C,Sutter G.Adiabatic shear banding in high speed
machining of Tie6Ale4V:experiments and modeling.International Journal of
Plasticity 2002;18:443e59.
[61] Khan AS,Kazmi R,Farrokh B,Zupan M.Effect of oxygen content and micro
structure on the thermomechanical response of three Tie6Ale4V alloys:
experiments and modeling over a wide range of strainrates and tempera
tures.International Journal of Plasticity 2007;23:1105e25.
[62] Suquet P.Overall properties of nonlinear composites:a modiﬁed secant
moduli theory and its link to Ponte Castañeda’s nonlinear variational proce
dure.C R Acad Sci Paris,Sèrie II b 1995;320:563e71.
[63] Suquet P.Overall properties of nonlinear composites:secant moduli theories
and variational bounds.In:Markov KZ,editor.Continuum models and
discrete systems,proceedings of 8th international symposium.World Scien
tiﬁc Publishing Company;1995.p.290e9.
[64] Suquet P.Overall properties of nonlinear composites:remarks on secant and
incremental formulations.In:Pineau A,Zaoui A,editors.Micromechanics of
plasticity and damage of multiphase materials (IUTAM symposium).Kluwer;
1996.p.149e56.
[65] ABAQUS/Explicit.Abaqus Explicit v6.10 user’s manual.Richmond,USA:
ABAQUS Inc.;2010.version 6.10 ed.
[66] Zukas JA,Scheffer DR.Practical aspects of numerical simulations of dynamic
events:effects of meshing.International Journal of Impact Engineering 2000;
24:925e45.
[67] RodríguezMartínez JA,Rusinek A,Arias A.Relation between strain hardening
of steel and critical impact velocity.Journal of Theoretical and Applied
Mechanics 2009;47:645e65.
[68] NeedlemanA.The effect of material inertia on neck development.In:Yang WH,
editor.Topics in plasticity.Ann Arbor,MI:AMPress;1991.p.151e60.
[69] RodríguezMartínez JA,Vadillo G,FernándezSáez J,Molinari A.Identiﬁcation
of the critical wavelength responsible for the fragmentation of ductile rings
expanding at very high strain rates,submitted for publication.
[70] Besnard G,Hild F,Lagrange JM,Martinuzzi P,Roux S.Analysis of necking in
high speed experiments by stereocorrelation.International Journal of Impact
Engineering 2012;49:179e91.
[71] Mott NF.Fragmentation of shell cases.Proceedings of the Royal Society of
London,Series A 1947:300e8.
[72] Grady D.Fragmentation of solids under impulsive stress loading.Journal of
Geophysical Research 1981;86:1047e54.
[73] Kipp M,Grady D.Dynamic fracture growth and interaction in one dimension.
Journal of the Mechanics and Physics of Solids 1985;33:399e415.
[74] Grady DE,Olsen ML.A statistics and energy based theory of dynamic frag
mentation.International Journal of Impact Engineering 2003;29:293e306.
J.A.RodríguezMartínez et al./International Journal of Impact Engineering 54 (2013) 206e216216
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